scholarly journals Lozenge Tilings of Hexagons with Central Holes and Dents

10.37236/8716 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Tri Lai
Keyword(s):  
Product Formula ◽  
Classical Formula ◽  
Plane Partitions ◽  
Equilateral Triangles ◽  
Lozenge Tilings ◽  
Simple Product ◽  
A Chain ◽  
Two Sides

Ciucu proved a simple product formula for the tiling number of a hexagon in which a chain of equilateral triangles of alternating orientations, called a `fern', has been removed from the center (Adv. Math. 2017). In this paper, we present a multi-parameter generalization of this work by giving an explicit tiling enumeration for a hexagon with three ferns removed, besides the central fern as in Ciucu's region, we remove two new ferns from two sides of the hexagon. Our result also implies a new `dual' of MacMahon's classical formula of boxed plane partitions, corresponding to the exterior of the union of three disjoint concave polygons obtained by turning 120 degrees after drawing each side.  

scholarly journals Enumeration of Hybrid Domino-Lozenge Tilings II: Quasi-Octagonal Regions

10.37236/4669 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Tri Lai
Keyword(s):  
Product Formula ◽  
Cambridge University ◽  
Replacement Method ◽  
Lozenge Tilings ◽  
Simple Product ◽  
Geometric Combinatorics

We use the subgraph replacement method to prove a simple product formula for the tilings of an  8-vertex counterpart of Propp's quasi-hexagons (Problem 16 in New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999), called quasi-octagon.

scholarly journals Lozenge Tiling Function Ratios for Hexagons with Dents on Two Sides

10.37236/9363 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Daniel Condon
Keyword(s):  
Triangular Lattice ◽  
Lozenge Tilings ◽  
Simple Product ◽  
Two Sides ◽  
Product Formulas

We give a formula for the number of lozenge tilings of a hexagon on the triangular lattice with unit triangles removed from arbitrary positions along two non-adjacent, non-opposite sides. Our formula implies that for certain families of such regions, the ratios of their numbers of tilings are given by simple product formulas.

scholarly journals A Simple Product Formula for Certain Kazhdan-LusztigR-Polynomials

2006 ◽  
Vol 34 (2) ◽  
pp. 407-414 ◽  
Author(s):  
Michela Pagliacci
Keyword(s):  
Product Formula ◽  
Simple Product

Mendelssohn's "Scottish" Tonality?

2006 ◽  
Vol 29 (3) ◽  
pp. 240-260
Author(s):  
Balazs Mikusi
Keyword(s):  
Large Scale ◽  
Sonata Form ◽  
Tonal Structure ◽  
Minor Mode ◽  
Major Mode ◽  
A Chain ◽  
The Relationship ◽  
Two Sides

Several of Mendelssohn's minor-mode songs, duets, and choral songs feature a peculiar tonal move: a sudden shift takes us to the relative major (without a "modulation" proper), but the opening minor key soon returns equally abruptly (via its V). Closer examination of these pieces suggests that the composer used the major-mode excursus as a topos, whose associations include the ideas of farewell, wandering, and distance (the latter both in the geographical and chronological sense, in accordance with the shift's quasi-modal--thus equally exotic and archaic--character). I suggest that this topos may have influenced the tonal structure of at least three large-scale Mendelssohn compositions, all of which are closely related to the same exotic and historical ideas. In the Hebrides Overture the relationship between the primary B minor and the secondary D major is (for a sonata-form movement) exceptionally equal: rather than acting as sharply contrasting tonal areas, they almost appear as two sides of the same key. The first-act finale of the unfinished opera, Die Lorelei, elaborates the original topos in another way: the E-minor-G-major kernel is extended in both directions, resulting in a chain of third-related keys, which eventually takes us back to the opening E level (now turned into major). In the light of this example, the (less complete) third-layered tonal structure of the "Scottish" Symphony may also be understood as growing out from the same miniature song topos.

scholarly journals Cyclic Sieving for Plane Partitions and Symmetry

2020 ◽  
Author(s):  
Sam Hopkins ◽  
Keyword(s):  
Fixed Points ◽  
Generating Function ◽  
Cyclic Group ◽  
Product Formula ◽  
Roots Of Unity ◽  
Plane Partitions ◽  
Cyclic Sieving Phenomenon ◽  
Combinatorial Set ◽  
Symmetric Plane ◽  
Cyclic Sieving

The cyclic sieving phenomenon of Reiner, Stanton, and White says that we can often count the fixed points of elements of a cyclic group acting on a combinatorial set by plugging roots of unity into a polynomial related to this set. One of the most impressive instances of the cyclic sieving phenomenon is a theorem of Rhoades asserting that the set of plane partitions in a rectangular box under the action of promotion exhibits cyclic sieving. In Rhoades's result the sieving polynomial is the size generating function for these plane partitions, which has a well-known product formula due to MacMahon. We extend Rhoades's result by also considering symmetries of plane partitions: specifically, complementation and transposition. The relevant polynomial here is the size generating function for symmetric plane partitions, whose product formula was conjectured by MacMahon and proved by Andrews and Macdonald. Finally, we explain how these symmetry results also apply to the rowmotion operator on plane partitions, which is closely related to promotion.

scholarly journals Signed Lozenge Tilings

10.37236/5579 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
D. Cook II ◽  
Uwe Nagel
Keyword(s):  
Perfect Matchings ◽  
Lattice Paths ◽  
New Method ◽  
Plane Partitions ◽  
Triangular Region ◽  
Lozenge Tilings

It is well-known that plane partitions, lozenge tilings of a hexagon, perfect matchings on a honeycomb graph, and families of non-intersecting lattice paths in a hexagon are all in bijection. In this work we consider regions that are more general than hexagons. They are obtained by further removing upward-pointing triangles. We call the resulting shapes triangular regions. We establish signed versions of the latter three bijections for triangular regions. We first investigate the tileability of triangular regions by lozenges. Then we use perfect matchings and families of non-intersecting lattice paths to define two signs of a lozenge tiling. Using a new method that we call resolution of a puncture, we show that the two signs are in fact equivalent. As a consequence, we obtain the equality of determinants, up to sign, that enumerate signed perfect matchings and signed families of lattice paths of a triangular region, respectively. We also describe triangular regions, for which the signed enumerations agree with the unsigned enumerations.

scholarly journals A triple product formula for plane partitions derived from biorthogonal polynomials

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Shuhei Kamioka
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Laguerre Polynomials ◽  
Product Formula ◽  
Triple Product ◽  
Lattice Paths ◽  
Plane Partitions ◽  
International Audience ◽  
Biorthogonal Polynomials ◽  
Bounded Size

International audience A new triple product formulae for plane partitions with bounded size of parts is derived from a combinato- rial interpretation of biorthogonal polynomials in terms of lattice paths. Biorthogonal polynomials which generalize the little q-Laguerre polynomials are introduced to derive a new triple product formula which recovers the classical generating function in a triple product by MacMahon and generalizes the trace-type generating functions in double products by Stanley and Gansner.

scholarly journals New Aspects of Regions whose Tilings are Enumerated by Perfect Powers

10.37236/3186 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Tri Lai
Keyword(s):  
Triangular Lattice ◽  
Product Formula ◽  
Square Lattice ◽  
Reduction Theorem ◽  
Simple Product ◽  
By Products

In 2003, Ciucu presented a unified way to enumerate tilings of lattice regions by using a certain Reduction Theorem (J. Algebraic Combin., 2003). In this paper we continue this line of work by investigating new families of lattice regions whose tilings are enumerated by perfect powers or products of several perfect powers. We prove a multi-parameter generalization of Bo-Yin Yang's theorem on fortresses (Ph.D. thesis, MIT, 1991).  On the square lattice with zigzag paths, we consider two particular families of regions whose numbers of tilings are always a power of 3 or twice a power of 3. The latter result provides a new proof for a conjecture of Matt Blum first proved by Ciucu. We also consider several new lattices obtained by periodically applying two simple subgraph replacement rules to the square lattice. On some of those lattices, we get new families of regions whose numbers of tilings  are given by products of several perfect powers. In addition, we prove a simple product formula for the number of tilings of a certain family of regions on a variant of the triangular lattice.

Syntheses of Two Co(II) and Ni(II) Complexes with 2-(1H-Imidazolyl-1- methyl)-1H-benzimidazole and 1,4-Benzenedicarboxylate Ligands and Their Effect on Cardiac Fibroblasts Proliferation

2013 ◽  
Vol 68 (11) ◽  
pp. 1225-1232 ◽  
Author(s):  
Ying Wang ◽  
Shuxun Yan ◽  
Yanzhou Zhang ◽  
Lina Sun ◽  
Zihan Wei
Keyword(s):  
Metal Ions ◽  
Cardiac Fibroblasts ◽  
Chain Structure ◽  
Cell Cycle Distribution ◽  
Obvious Effect ◽  
Flow Cytometric ◽  
A Chain ◽  
Vitro Effect ◽  
Two Sides

Two new isostructural complexes based on 2-(1H-imidazolyl-1-methyl)-1H-benzimidazole (imb) and di-anionic 1,4-benzenedicarboxylate (bdic), namely, {[Co(bdic)(imb)2(H2O)2]·2H2O}n (1) and {[Ni(bdic)(imb)2(H2O)2]·2H2O}n (2), have been synthesized and characterized by single-crystal Xray diffraction. Both complexes possess a chain structure with the di-anionic bdic groups bridging the metal ions. The imb ligands coordinate the metal ions in a monodentate mode at two sides of the main chain. These chains are further packed into 3D networks through five kinds of hydrogen bonds. The in vitro effect of 1 and 2 on cultured cardiac fibroblasts (CF) proliferation in the presence or absence of excessive angiotensin II (AngII) have been investigated by a flow cytometric assay. The results have indicated that both complexes have no obvious effect on the cell cycle distribution of CF, but they can suppress the CF proliferation induced by AngII.

scholarly journals The Number of Optimal Matchings for Euclidean Assignment on the Line

2021 ◽  
Vol 183 (1) ◽  
Author(s):  
Sergio Caracciolo ◽  
Vittorio Erba ◽  
Andrea Sportiello
Keyword(s):  
Generating Functions ◽  
Arbitrary Order ◽  
Random Variable ◽  
Singularity Analysis ◽  
Product Formula ◽  
Zero Temperature ◽  
Brownian Process ◽  
The Mean ◽  
Simple Product ◽  
Linear Cost

AbstractWe consider the Random Euclidean Assignment Problem in dimension $$d=1$$ d = 1 , with linear cost function. In this version of the problem, in general, there is a large degeneracy of the ground state, i.e. there are many different optimal matchings (say, $$\sim \exp (S_N)$$ ∼ exp ( S N ) at size N). We characterize all possible optimal matchings of a given instance of the problem, and we give a simple product formula for their number. Then, we study the probability distribution of $$S_N$$ S N (the zero-temperature entropy of the model), in the uniform random ensemble. We find that, for large N, $$S_N \sim \frac{1}{2} N \log N + N s + {\mathcal {O}}\left( \log N \right) $$ S N ∼ 1 2 N log N + N s + O log N , where s is a random variable whose distribution p(s) does not depend on N. We give expressions for the moments of p(s), both from a formulation as a Brownian process, and via singularity analysis of the generating functions associated to $$S_N$$ S N . The latter approach provides a combinatorial framework that allows to compute an asymptotic expansion to arbitrary order in 1/N for the mean and the variance of $$S_N$$ S N .

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